New additive-consistency-driven methods for deriving two types of normalized utility vectors from additive reciprocal preference relations

In multi-criteria decision making, importance weights of criteria are usually derived from pairwise comparisons and requested for adding up to one. The framework of additive reciprocal preference relations (ARPRs) is a commonly used tool representing pairwise comparisons under the unit interval scale. This paper illustrates that existing additive-consistency-driven utility derivation methods fail to acquire a satisfactory additively normalized importance weight vector of criteria. A normalization framework is introduced under an Abelian linearly ordered group and new methods are proposed to generate additively consistent ARPRs respectively from vectors satisfying new normalization conditions and additively normalized vectors. Two formulas are devised to obtain the two types of normalized utility vectors associated with an additively consistent ARPR. Three optimization models are built and their analytical solutions are found to derive the two types of normalized optimal utility vectors from ARPRs. A geometrically additive consistency index is presented and its approximated thresholds are suggested to check acceptably additive consistency of ARPRs. A stepwise procedure is developed to deal with multi-criteria decision making problems. Two numerical illustrations with comparative analysis and an international exchange postgraduate student selection problem are offered to demonstrate the performance and practicality of the presented models.